I am going to be pedantic in this problem and do the full derivation for the finding the bn coefficients because this stuff is really confusing and I think explaining the steps is valuable. We know we want some bn such that f (x) = X n=0 bn sin 2 P nx (1) where P is the length of the period of f (x). Since we can only build an odd function from these sine functions because they are odd themselves, we can already tell we'll need to do some legwork to approximate the even function f (x) = 1 x2 L2 . For now, we can continue naively and simply deal with it later. Equation (1) is great but not particularly helpful to pin down bn. Our main challenge is dealing with the sum. One tool we can use is orthogonality. Specifically, we can use that R sin 2 P mx sin 2 P nx dx = 0 for m = n to make almost all of the terms in the sum disappear and get a much simpler expression to manipulate to find bn